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36 votes
1 answer
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Is there a general theory of "compactification"?

In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
Tim Campion's user avatar
  • 63.9k
12 votes
1 answer
1k views

Reference Request for Drinfeld and Laumon Compactifications

Background Let $X$ denote a smooth projective curve over $\mathbb{C}$ and let $G$ denote a semi-simple simply connected algebraic group over $\mathbb{C},$ which has associated flag variety $G/B.$ ...
Mike Skirvin's user avatar
  • 2,706
10 votes
2 answers
599 views

Non-bimeromorphic compactifications

Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify ...
diverietti's user avatar
  • 7,902
9 votes
1 answer
739 views

Compactifications of varieties with small complement

Let $X$ be a smooth variety over an algebraically closed field $k$. If it makes things easier, $X$ may be assumed to be quasi-projective. By Nagata (or quasi-projectivity) there exists a proper ...
Lars's user avatar
  • 4,450
8 votes
1 answer
1k views

Wonderful compactification

Suppose $G$ is a semi-simple group of adjoint type over an algebraic closed field, and $X$ its wonderful compactification a la De Concini and Procesi. Let $P=MU$ be a parabolic subgroup in $G$, and ...
Ramin's user avatar
  • 1,362
8 votes
1 answer
439 views

Does a flat compactification always exist?

Let $\pi:X\to S$ be a separated flat morphism of finite type of Noetherian schemes. Does $\pi$ necessarily factor as an open immersion followed by a proper flat morphism? The analogue of this question ...
user avatar
8 votes
0 answers
522 views

Deligne-Mumford moduli spaces and compactification of symmetric matrices

The real Deligne-Mumford moduli space $\bar M_{0,n+1}(\mathbb R)$ of stable genus zero curves with $n+1$ marked points is a compactification of the space of configurations of $n$ distinct ordered ...
Konrad Schöbel's user avatar
7 votes
1 answer
578 views

Flatly compactifiable morphisms

Let $f:U \to S$ be a flat morphism. Let us say that $f$ is flatly compactifiable if there exists a proper morphism $\bar{f}:X \to S$ and a closed subscheme $Z \subset X$ such that 1) $U = X \...
Sasha's user avatar
  • 39.3k
7 votes
0 answers
499 views

Compactification of the moduli space of Kähler manifolds with negative constant scalar curvatures

Moishezon compactification is very important in the study of the moduli space of varieties which admit canonical metrics. Moishezon showed that any non-projective Moishezon manifold $X$, after a ...
user avatar
7 votes
0 answers
491 views

Alterations of regular varieties

Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
Lars's user avatar
  • 4,450
6 votes
0 answers
159 views

Completion/Compactification of a Kähler metric on $\mathbb C^2$

Consider $\mathbb{C}^{2}$ equipped with the Kähler form $$ \omega_{\mu}=\frac{i}{2} \partial \bar{\partial} \log \left(\left(1+|z|^{2}\right)^{\mu}+|w|^{2}\right), $$ where $\mu$ is a positive real ...
Robbixmaths's user avatar
5 votes
0 answers
273 views

Intuition for the McGerty-Nevins compactification of quiver varieties

In Section 4 of the paper Kirwan surjectivity for quiver varieties (Inventiones Math. 2018) McGerty and Nevins define a compactification of the moduli space of representations of the preprojective ...
Yellow Pig's user avatar
  • 2,964
5 votes
0 answers
294 views

On toroidal compactifications of Hilbert Kuga-Sato varieties

Let $F$ be a totally real field of degree d. There are Hilbert modular varieties over $\mathbb{Q}$ that paramatrize abelian varieties of dimension d with an action of $\mathcal{O}_F$ the ring of ...
Bear's user avatar
  • 845
4 votes
2 answers
866 views

Non-uniqueness of smooth compactification

Let $U$ be a smooth quasi-projective variety. Does there always exist a smooth compactification of $U$? If not always when can we have smooth compactification? In particular, suppose $X$ is a ...
Naga Venkata's user avatar
  • 1,040
4 votes
1 answer
531 views

Uniqueness of smooth compactification upto a smooth morphism

By a $k$-variety, we will mean a separated scheme of finte type over a field $k$. Let $k$ be of characteristic 0. Given a smooth quasi-projective $k$-variety $X$, there is a projective $k$-variety $\...
Anandam Banerjee's user avatar
4 votes
1 answer
378 views

Minimal Nagata-like compactification

Working over a field $k$, Nagata's compactification theorem implies that any separated scheme $X$ of finite type over $k$ admits a compactification (a dense open immersion $i \colon X\hookrightarrow\...
user337830's user avatar
4 votes
0 answers
487 views

Embedding of a smooth variety into a complete smooth variety.

Consider the following fact from algebraic geometry: Any (complex) smooth algebraic variety can be embedded into a complete smooth variety as a locally closed set. I know how to prove this fact ...
Rami's user avatar
  • 2,639
3 votes
0 answers
107 views

Fulton-MacPherson compactifications (and wonderful compactifications) as relative Proj

Let $X$ be a (smooth complex projective) variety. The Fulton-MacPherson compactification $X[n]$ is obtained from $X^n$ by blowing up the diagonals in a certain order. Is it possible to write down a (...
adrian's user avatar
  • 318
3 votes
0 answers
204 views

Smooth toric compactification of $\mathbb C^n$

By a compactification $(X,Y)$ of $\mathbb C^n$, we mean an irreducible compact complex space $X$ and a closed analytic subspace $Y\subset X$ such that $X\setminus Y$ is biholomorphic to $\mathbb C^n$. ...
Hang's user avatar
  • 2,789
3 votes
0 answers
195 views

Universal closure of schemes à la Nagata

Nagata compactification theorem is the following fundamental result: Let $S$ be a qcqs scheme. Let $X$ be a separated $S$-scheme of finite type. Then there exists a proper $S$-scheme $\overline{X}$ ...
user avatar
3 votes
0 answers
443 views

Equivariant form of Nagata's compactification theorem?

Given a finite group $G$ acting on an algebraic variety $X$ (let's say over $\mathbb C$, if that helps), is there always a proper variety $\bar X$ with a $G$ action such that $X \to \bar X$ is a $G$-...
Dominik's user avatar
  • 3,017
2 votes
1 answer
123 views

Naive compactification of $\mathbb{C}^*$-fibrations

Let $\pi:X \to Y$ be a $\mathbb{C}^*$-fibration between complex manifolds in the sense that there exists a fixed integer $a$ such that for every $y \in Y$, $\pi^{-1}(y)=(\mathbb{C}^*)^a$. Suppose ...
Jana's user avatar
  • 2,022
2 votes
1 answer
180 views

Locally toric resolutions of compactifications

Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an ...
Dmitry Vaintrob's user avatar
2 votes
0 answers
102 views

What is the meaning of universal family of Fulton Macpherson configuration space?

Fulton and Macpherson suggests the way to compactify the set of $n$-labelled distinct point on variety in their paper, "A Compactification of Configuration Spaces" In this paper, the process ...
ChoMedit's user avatar
  • 285
2 votes
0 answers
152 views

Line bundles on toric varieties associated to Weyl chamber

I am interested in studying toric varieties associated to the fan of Weyl chambers. General information would be best but I am also interested in the specific case of the Weyl chamber of $\mathfrak{sl}...
Merrick Cai's user avatar
2 votes
0 answers
218 views

Borel-Weil-Bott theorem for wonderful compactification in characteristic p

Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
Merrick Cai's user avatar
2 votes
0 answers
159 views

toroidal compactifications of modulis spaces of ppav's

Are the modular toroidal compactifications of ppav's (second Voronoi) defined by Alexeev without self-intersections? i.e. are the irreducible component of the boundary divisor normal? If not, can one ...
cata's user avatar
  • 357
2 votes
0 answers
191 views

degenerate abelian surfaces

I am wondering if the family of degenerate abelian surfaces constructed by K. Hulek, C. Kahn and S.H. Weintraub in "Moduli spaces of Abelian Surfaces: Compactification, Degenarations, and Theta ...
cata's user avatar
  • 357
2 votes
0 answers
574 views

tangent bundle of the toric variety of the wonderful compactification.

Let G be a adjoint group over $k$,algebraically closed of caracteristic zero. Let $\overline{G}$ be its wonderful compactification. I denote by $\overline{T}$ the closure of the torus $T$ in $\...
prochet's user avatar
  • 3,472
1 vote
0 answers
263 views

Implicit function theorem and compactification of algebraic curve

Let $C$ be a singular curve defined over a local field $K$. Let $\tilde{C}$ be its smooth compactification(maybe this is not normalization). Why $\tilde{C}(K)\neq \emptyset$ implies ${C}(K)\neq \...
Duality's user avatar
  • 1,531
1 vote
0 answers
501 views

The Deligne-Mumford Compactification for Closed Surfaces

I am reading this note on super-Riemann surfaces. In the second paragraph of section 7.4.1 (page 87), there is a statement that I am trying to understand: The compactified moduli space of closed ...
QGravity's user avatar
  • 989
1 vote
0 answers
68 views

Extending a model to a given compactification of its generic fiber

Let $R$ be a discrete valuation ring and $K$ its field of fraction. Let $X$ be a proper $K$-variety, $U$ a dense open and consider an $R$-model $\mathcal{U}$ of $U$. Can we embed $\mathcal{U}$ in a ...
Piet's user avatar
  • 11
0 votes
1 answer
204 views

Are degrees and ramification degrees preserved upon passing to the smooth compactification?

Let $\phi :C_1\to C_2$ be morphism of projective singular curve. Let $\tilde{C}_1$ and $\tilde{C}_2$ be their smooth compactification. Then $\phi$ extends to $\tilde{\phi} : \tilde{C}_1\to \tilde{C}...
Duality's user avatar
  • 1,531